[Paper Review] 4 A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL p–LAPLACE EQUATIONS

Jérôme Vétois|arXiv (Cornell University)|Aug 17, 2016

Nonlinear Partial Differential Equations36 references81 citations

TL;DR

This paper establishes sharp pointwise a priori estimates for solutions to critical p-Laplace equations in R^n, proving decay rates for both solutions and their gradients. By combining weak Lebesgue space bounds, rescaling techniques, and Harnack-type inequalities, the authors derive explicit decay rates that extend radial symmetry results to the full range 1 < p < 2 for positive solutions of −Δ_p u = u^{p^*-1}, confirming that the known radial solutions are the only positive solutions in D^{1,p}(R^n).

ABSTRACT

We establish pointwise a priori estimates for solutions in $D^{1,p}(\mathbb{R}^n)$ of equations of type $-\Delta_pu=f(x,u)$, where $p\in(1,n)$, $\Delta_p:=\mbox{div}\big(\left| abla u ight|^{p-2} abla u\big)$ is the $p$-Laplace operator, and $f$ is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy, we are able to extend a recent result of Damascelli-Merch\'an-Montoro-Sciunzi on the symmetry of positive solutions in $D^{1,p}(\mathbb{R}^n)$ of the equation $-\Delta_pu=u^{p^*-1}$, where $p^*:=np/(n-p)$.

Motivation & Objective

• Establish pointwise a priori estimates for solutions in D^{1,p}(R^n) of critical p-Laplace equations with nonlinearity f(x,u) satisfying critical Sobolev growth.
• Extend radial symmetry results for positive solutions beyond the previously known range 2n/(n+2) ≤ p < 2 to the full interval 1 < p < 2.
• Prove that the only positive solutions in D^{1,p}(R^n) of −Δ_p u = u^{p^*-1} are the known radial, symmetric solutions of Guedda–Véron.
• Provide a priori decay estimates that are crucial for symmetry proofs via moving plane methods.
• Supply refined decay estimates (upper and lower bounds) that are sharp in scaling and essential for global compactness arguments.

Proposed method

• Derive global L^{p^*-1,∞} bounds for solutions using measure-theoretic arguments and truncation techniques on the weak formulation.
• Apply rescaling to the equation to analyze asymptotic behavior at infinity, transforming the problem into a limiting one on the whole space.
• Use the doubling property of Poláčik–Quittner–Souplet to control the growth of solutions in annular regions.
• Establish a Harnack inequality on annuli via Serrin’s and Trudinger’s results, enabling comparison of sup and inf over large annuli.
• Combine the rescaling argument with Harnack estimates and interpolation to derive sharp upper bounds on |u(x)| and |∇u(x)|.
• Prove a lower bound on positive solutions by combining a Harnack inequality on annuli with a lower bound on the energy integral ∫ f(x,u)dx.

Experimental results

Research questions

• RQ1 What are the sharp pointwise decay estimates for solutions of critical p-Laplace equations in R^n with critical nonlinearity?
• RQ2 How can a priori estimates be used to extend radial symmetry results to the full range 1 < p < 2 for positive solutions?
• RQ3 What is the precise asymptotic behavior of solutions to −Δ_p u = u^{p^*-1} in D^{1,p}(R^n)?
• RQ4 Can the decay rates of solutions and gradients be quantified in terms of the scaling of the equation?
• RQ5 Under what conditions does the Harnack inequality on annuli yield a lower bound on positive solutions?

Key findings

• The solution u satisfies the pointwise upper bound |u(x)| ≤ C₀ (1 + |x|)^{-(n−p)/(p−1)} and |∇u(x)| ≤ C₀ (1 + |x|)^{-(n−1)/(p−1)} for all x ∈ R^n.
• The constant C₀ depends only on n, p, Λ, and norms of u in L^{p^*-1,∞}(R^n) and W^{1,∞}(B(0,3r)).
• For positive solutions with ∫_{R^n} f(x,u) dx > 0, a lower bound u(x) ≥ C₁ (1 + |x|)^{-(n−p)/(p−1)} holds, with C₁ depending on n, p, λ, Λ, and lower bounds of u.
• The Harnack inequality on annuli ensures that sup_{2R<|x|<5R} u(x) ≤ c₄ inf_{2R<|x|<5R} u(x) for all R ≥ r, with c₄ independent of R.
• By combining the a priori estimates with results of Damascelli–Ramaswamy and Li, it is shown that all positive D^{1,p}(R^n) solutions of −Δ_p u = u^{p^*-1} are radially symmetric and strictly decreasing.
• Corollary 1.3 confirms that the Guedda–Véron solutions (1.10) are the only positive solutions in D^{1,p}(R^n) for 1 < p < 2.